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We discuss a Gaussian multiplicative chaos (GMC) structure underlying afamily of random measures $\mathbf{M}_r$, indexed by $r\in\mathbb{R}$, on aspace $\Gamma$ of directed pathways crossing a diamond fractal with Hausdorffdimension two. The laws of these random continuum path measures arise in acritical weak-disorder limiting regime for discrete directed polymers ondisordered hierarchical graphs. For the analogous subcritical continuum polymermodel in which the diamond fractal has Hausdorff dimension less than two, therandom path measures can be constructed as subcritical GMCs through couplingsto a spatial Gaussian white noise. This construction fails in the criticaldimension two where, formally, an infinite coupling strength to theenvironmental noise would be required to generate the disorder. We prove,however, that there is a conditional GMC interrelationship between the randommeasures $(\mathbf{M}_r)_{r\in \mathbb{R}}$ such that the law of $\mathbf{M}_r$can be constructed as a subcritical GMC with random reference measure$\mathbf{M}_R$ for any choice of $R\in (-\infty, r)$. A similar GMC structureplausibly would hold for a critical continuum (2+1)-dimensional directedpolymer model.

The conditional Gaussian multiplicative chaos structure underlying a critical continuum random polymer model on a diamond fractal